Question

Find the indefinite integral in two ways. Explain any difference in the forms of the answers. $$ \int \sin x \cos x d x $$

Short answer

The indefinite integral \( \int \sin x \cos x \, dx \) can be solved using two substitutions \( u = \sin x \) and \( u = \cos x \) to obtain two forms of the answer: \( \frac{1}{2}\sin^2 x + C \) and \( -\frac{1}{2} + \frac{1}{2}\sin^2 x + C \), all of which represents the same amount, but in different ways.

Step-by-step solution

Solving using substitute \( u = \sin x \)

Start by letting \( u = \sin x \). Then, differentiate \( u \) to find \( du \). Using the fact that the derivative of \( \sin x \) is \( \cos x \), obtain \( du = \cos x \, dx \). So, the integral then becomes \( \int u \, du \) which is easy to solve.

Calculating the integral

The antiderivative of \( u \) with respect to \( u \) is \( \frac{1}{2}u^2 \). So, \( \int u \, du = \frac{1}{2}u^2 + C \), where \( C \) is the constant of integration.

Reverse substitution

Substitute back \( u = \sin x \) into the result from Step 2 to get \( \frac{1}{2}\sin^2 x + C \). This is the result derived from the first method.

Solving by substituting \( u = \cos x \)

Now let's use a different substitution: let \( u = \cos x \). Then, differentiate \( u \) to find \( du \), which gives \( du = -\sin x \, dx \). Rearranging, we get \( -du = \sin x \, dx \). So, the integral becomes \( - \int u \, du \) which is again easy to solve.

Computing the integral

The antiderivative of \( u \) with respect to \( u \) is \( \frac{1}{2}u^2 \). So, \( - \int u \, du = -\frac{1}{2}u^2 + C = -\frac{1}{2}\cos^2 x + C \), where \( C \) is the constant of integration. This is the result derived from the second method.

Mason's identity to explain equivalence

Let's use the double angle formula (also known as Cosine square identity), which states \( \cos^2 x = 1 - \sin^2 x \). Applying this identity to the result from Step 5, we obtain \( -\frac{1}{2}(1 - \sin^2 x) + C = -\frac{1}{2} + \frac{1}{2}\sin^2 x + C \). This form is equivalent to the result in Step 3, just in a different form.